Problem 4.19 A bead slides down a wire. You want to determine the shape the wire must have for the bead to slide from its initial position to its final position in the least amount of time. (This is called the "brachistochrone" problem.) Note that the quantity to be minimized is the time and that dt = ds/v where the velocity v is given by conservation of energy. I do not expect you to actually solve this problem; it will be sufficient if you write the functional that describes the system.12

Problem 4.19 A bead slides down a wire. You want to determine the shape the wire must have for the bead to slide from its initial position to its final position in the least amount of time. (This is called the "brachistochrone" problem.) Note that the quantity to be minimized is the time and that dt = ds/v where the velocity v is given by conservation of energy. I do not expect you to actually solve this problem; it will be sufficient if you write the functional that describes the system.12